http://informahealthcare.com/ahb ISSN: 0301-4460 (print), 1464-5033 (electronic) Ann Hum Biol, Early Online: 1–8 ! 2015 Informa UK Ltd. DOI: 10.3109/03014460.2015.1046487
RESEARCH PAPER
Combining wrist age and third molars in forensic age estimation: how to calculate the joint age estimate and its error rate in age diagnostics* Bianca Gelbrich1, Carolin Frerking1, Sandra Weiß2, Sebastian Schwerdt1, Angelika Stellzig-Eisenhauer2, Eve Tausche3, and Go¨tz Gelbrich4,5 1
Department of Orthodontics, University Hospital Leipzig, Germany, 2Department of Orthodontics, University Hospital Wu¨rzburg, Germany, Department of Orthodontics, University Hospital Dresden, Germany, 4Institute of Clinical Epidemiology and Biometry, University of Wu¨rzburg, Germany, and 5Clinical Trial Centre, University Hospital Wu¨rzburg, Germany
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Abstract
Keywords
Background: Forensic age estimation in living adolescents is based on several methods, e.g. the assessment of skeletal and dental maturation. Combination of several methods is mandatory, since age estimates from a single method are too imprecise due to biological variability. The correlation of the errors of the methods being combined must be known to calculate the precision of combined age estimates. Aim: To examine the correlation of the errors of the hand and the third molar method and to demonstrate how to calculate the combined age estimate. Subjects and methods: Clinical routine radiographs of the hand and dental panoramic images of 383 patients (aged 7.8–19.1 years, 56% female) were assessed. Results: Lack of correlation (r ¼ –0.024, 95% CI ¼ –0.124 to + 0.076, p ¼ 0.64) allows calculating the combined age estimate as the weighted average of the estimates from hand bones and third molars. Combination improved the standard deviations of errors (hand ¼ 0.97, teeth ¼ 1.35 years) to 0.79 years. Conclusion: Uncorrelated errors of the age estimates obtained from both methods allow straightforward determination of the common estimate and its variance. This is also possible when reference data for the hand and the third molar method are established independently from each other, using different samples.
Age estimation error, combined methods, dental panoramic radiograph, forensic age estimation, hand radiograph
Introduction The application of a law may depend on the age of the subject concerned. In Germany, for example, the age of criminal responsibility is 14 years. Criminal law relating to young offenders must be applied if the delinquent has not yet reached the age of 18 years and can be applied if the delinquent is still under the age of 21. Beyond criminal proceedings, young immigrants under 18 years of age deserve particular protection by the German law. The same or similar rules are applied in many other countries. In cases where age-dependent legal action is due and the age of the subject under consideration cannot be verified (for example if a young immigrant does not possess identifying documents), the authority needs to assume an age. The assumption is made on the basis of forensic expertise. Forensic age estimation explores the maturation stage of several biological structures. According to current
This paper is dedicated to the memory of Klaus Ro¨tzscher. Correspondence: Dr. Bianca Gelbrich, Department of Orthodontics, University Hospital Leipzig, Liebigstraße 10-14, Building 1, D-04103 Leipzig, Germany. Tel: +49-341-97 21 050. Fax: +49-341-97 21 059. E-mail: [email protected]
History Received 23 April 2015 Accepted 27 April 2015 Published online 16 June 2015
recommendations (Schmeling et al., 2008), the development of the hand assessed from a radiograph, mineralisation of third molars assessed from a panoramic image and the closure of the clavicular epiphysis assessed from computed tomography should be incorporated when the age of living individuals is estimated in the context of criminal proceedings. Apart from the latter, German law allows X-ray imaging only in the presence of a clinical indication. Depending on the country and national law, the forensic methods may then be limited to assessing the eruption of the third molars or, in rare cases, exploring a single radiograph when available from a medical treatment. The timing of bone calcification and mineralisation of teeth shows large biological variability. Developmental stages of subjects of the same age may be very different and, vice versa, individuals with identical mineralisation of a certain bone or tooth may differ in age by 4 years or more. As a consequence, the estimation errors (differences between estimated and true chronological ages) have a substantial variance when age would be estimated from a single biological characteristic and age diagnostics (i.e. the decision whether a certain age limit has been passed) would suffer from unacceptable error rates. Therefore, several biological structures need to be assessed and the information from
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different estimation methods should be weighed against each other and summarised in the forensic expertise. In common practice separate age estimates and measures of uncertainty (e.g. standard deviations) are quoted for each method and the overall estimate of the most probable age and statements with regard to the age limits relevant in the respective legal context are derived by discussion rather than computation. There is no established strict procedure for the aggregation of the particular estimates and the determination of the variance of the pooled age estimate. The latter would be of particular interest as it determines the precision of forensic age diagnostics. Consider two methods of age estimation, suppose their errors have equal variances, denoted by V, and let r be the correlation coefficient of the errors of both methods. Then the errors of the average of the two age estimates has the variance V (1 + r)/2. If, for example, the left and the right hand would be examined, one can assume that the skeletal age is accelerated or delayed for both hands equally or almost equally, so r would be close to 1, and the variance of the averaged age estimates would be almost as large as V. This means that, given the information from one hand, the examination of the other does not add much and does not materially contribute in improving the precision of the estimated age. If, in another scenario, acceleration or delay of the development of two biological characteristics were independent from each other, the resulting two estimates of age would have uncorrelated errors. This would be the ideal case since each method would add independent information to the other and the variance of the averaged age estimates would be equal to V/2. For the common interpretation of age estimates from different methods it is, therefore, crucial to have information about the correlation between the estimation errors. The aim of this work is to study the correlation between the errors of the age estimates obtained from the hand bones and from the mineralisation of third molars.
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Methods
more precise information on age in younger children, mineralisation of this part of the permanent dentition is finished at the age of 14 years in about half of the population. Therefore, in the setting of forensic age estimation in living subjects within legal proceedings, incomplete mineralisation of these teeth would imply sufficient doubt that the subject under assessment was older than 14 years. On the other hand, complete mineralisation would not be used to derive any statement regarding the question of whether the subject has or has not passed the threshold of 14 years or another legally relevant age limit. Hence, only third molars are helpful in the diagnosis of, for example, criminal responsibility. The mineralisation stages of third molars were assessed according to the scheme of Demirjian et al. (1973). Eight stages are defined in this classification: calcified cusp tips (A), connected occlusal surface (B), complete formation of enamel in the occlusal surface (C), completion of crown formation (D), pulp horns and pulp chamber differentiated, radicular bifurcation (E), root length starts exceeding crown height, funnel shaped apices (F), root almost complete, apices still open (G) and apices closed (H). For computations, these stages were enumerated from 1–8. Following an idea from Nolla’s (1960) classification, we added the stage of a crypt without visible mineralisation, enumerated as 0. This enables better distinguishing between teeth which are present and will start mineralising soon and teeth that are not visible at all and might develop later or even never due to aplasia. The classification was carried out by the second author after training by the first. For quality assurance, the first author re-evaluated independently a randomly chosen part of the panoramic images. In addition, statistical monitoring was performed by the last author. For conversion of Demirjian stages into estimated ages, a regression model was applied (see statistics below). We did not use established formulae as we extended Demirjian’s scheme by the 0 stage. This stage was not included in the published assessment schemes.
Subjects and materials
Estimation of skeletal age
We retrospectively evaluated routine X-ray images of patients of the orthodontic departments of the University Hospitals of Leipzig, Dresden and Wu¨rzburg, Germany. Subjects were eligible if they had a radiograph of their hand and a panoramic dental radiograph in their files, the time difference between the dates of both images was not greater than 6 months, the development of at least one third molar was visible on the panoramic image and the subject had no syndrome known to be associated with a delay of the general, skeletal or dental maturation. Cases of both hypodontia, supernumerary teeth and preceding tooth extractions were also excluded. Radiographs were eligible if they had sufficient quality to perform the assessments described below. If more than one pair of radiographs was available from a subject, the one with the smaller difference of dates was selected. In cases of equality, the radiographs of better quality were preferred.
The age estimates from the hand X-rays were obtained by the atlas method of Thiemann et al. (2006). The atlas handbook contains a series of reference radiographs of the left hand, one for each age group and sex, accompanied by a description of the developmental features of the hand bones which are characteristic for the respective age. The observer should iteratively find the reference chart that best fits the image under assessment and this determines the estimated age. The assessment was carried out by the third and the fourth author (for the images from Leipzig/Dresden and Wu¨rzburg, respectively). Again, the first and last author provided training and supervision.
Estimation of dental age Dental age was estimated from the development of third molars. Although incisors through second molars provide
Statistics Estimated dental age was calculated by a multiple linear regression model, adjusting for sex and the structure of missing third molars. The estimation errors were computed as the differences between estimated ages and chronological age. The relationship between both estimation errors was assessed by Pearson’s correlation coefficient. Its 95% confidence
Combining methods of age estimation
DOI: 10.3109/03014460.2015.1046487
interval (CI) was computed using Fisher’s transform (Sachs, 1997, pp. 541–542). SPSS 22 (IBM Corp., Armonk, NY) was used as statistical software. Ethics Since there were no experimental issues, data were obtained retrospectively from clinical routine images and data transferred for analysis were anonymous, neither ethical approval nor informed consent of the subjects was necessary according to legal requirements in Germany. Nonetheless, the study protocol defining the research project, which the present work is part of, has been approved by the Ethics Committee of the Medical Faculty of Leipzig University, Germany.
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Sample characteristics The sample included 383 subjects, 213 girls (56%) and 170 boys. Ages ranged from 7.8–19.1 years. Frequencies of age groups and sexes are displayed in Table 1. The radiographs of 264 subjects (69%) were obtained within 1 month and in a further 80 subjects (21%) the dates of the radiographs were more than 1 month but no more than 3 months apart from each other. Estimated dental age The regression formula for age estimation from third molars, taking into account the configuration of present third molars and the sex of the subject, yielded: Estimated age ðyearsÞ ¼ 7:79 þ 1:26 average Demirjianstage of third molars þ 0:54 ½third molars incomplete AND ðN Max 4 N Mand Þ þ1:27 ½third molars incomplete AND ðN Max N Mand Þ 0:16 ½female sex where NMax and NMand denote the numbers of present maxillary and mandibular third molars, respectively. Each of the three logical expressions in brackets takes the numerical value 1 if true and 0 if false. The coefficients and their 95% CI were: intercept 7.79 (7.33–8.24), slope 1.26 (1.16–1.36), shifts for at least one missing third molar 0.54 (0.09–0.98) if NMax4NMand and 1.27 (0.87–1.66) if NMax NMand. The shift for female sex (0.16, 0.44 to +0.11) was not formally significant (p ¼ 0.25), but was left in the model since a slightly earlier development of third molars in girls (our estimate yields 2 months) is plausible. Table 1. Distribution of age and sex in the sample. Age group (years) 7.8–59 9–510 10–511 11–512 12–513 13–514 14–515 15–516 16–19.1
n (boys)
n (girls)
3 18 29 23 32 28 16 13 8
15 22 38 42 34 23 14 14 11
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Since this regression model was fitted to the data, there is no bias by construction. The standard deviation of the residuals, i.e. the estimation errors, was 1.35 years. The relationship between age estimates obtained from third molars and chronological age is illustrated in Figure 1(A). Estimated skeletal age The estimates of age from hand radiographs according to the Thiemann-Nitz method were slightly biased, the mean error was 0.37 years (95% CI ¼ –0.47 to 0.27). This corresponds to an under-estimation of age by 4 months, which is in favour of the subjects under forensic assessment. The standard deviation of the estimation errors was 0.97 years. The relationship between age estimates obtained from the hand and chronological age is illustrated in Figure 1(B). Correlation of estimation errors The correlation coefficient of the estimation errors of both methods was 0.024 (95% CI ¼ –0.124 to +0.076, p ¼ 0.64). Figure 2(A) illustrates the lack of correlation. Selecting only the 264 subjects whose radiographs were obtained within 1 month yielded a correlation coefficient of 0.007 (95% CI ¼ –0.128 to + 0.114, p ¼ 0.91). Looking at Figure 2, we can divide the errors of both methods into tertile groups. Let T1, T2, T3 and H1, H2, H3 denote the lowest, middle and highest tertile groups of the estimation errors of the third molar and hand method, respectively (Figure 2B). One might expect a positive association in the sense that, for example, when dental age is accelerated it would be more likely that the skeletal age is also accelerated rather than delayed. In other words, within the group T3 more subjects belonging to H3 than to H1 would have been expected. Vice versa, within T1 one would have expected an over-representation of H1 and an under-representation of H3. This was not the case. The lack of correlation is reflected by the nearly equal numbers of subjects in all nine combinations of tertiles. Aggregated estimated age The aggregated estimate of age Aa is computed as the weighted average of ages estimated from the third molars (AT) and the hand (AH): Aa ¼ wT AT þwH AH The respective weights wT and wH are non-negative numbers fulfilling wT + wH ¼ 1. We now derive how wT and wH should be chosen to minimise the variance of errors "a of the aggregated estimate. The errors are equal to "a ¼ wT AT þ wH AH Ac ¼ wT ðAc þ"T Þ þ wH :ðAc þ"H Þ Ac ¼ wT :"T þ wH :"H where Ac is the chronological age and "T, "H are the errors of the age estimates obtained from teeth and
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Figure 2. (A) Scatter plot of the errors of ages estimated from hand (horizontal axis) and teeth (vertical axis). The regression line illustrates the lack of a meaningful correlation. (B) Frequencies of subjects in the tertile groups of the estimation errors. The combinations of tertiles of both methods occur nearly equally often.
Figure 1. Ages estimated from third molars (A) and the hand (B) plotted against chronological age. In regard of the diagnosis of passing the age limit of 14 years, data points in the upper left quadrant defined by the dashed lines represent false negative cases and false positives are in the lower right quadrant.
hand, respectively. Thanks to the null correlation of the errors "T and "H and applying wH ¼ 1 – wT, the variance of "a computes as V½"a ¼ w2T V½"T þð1wT Þ2 V½"H ¼ ðV½"T þV½"H Þ w2T 2 V½"H wT þV½"H This quadratic function in wT attains its minimum at the zero of its derivative d=dwT V½"a ¼ 2 ðV½"T þ V½"H ÞwT 2 V½"H ¼ 0 which implies wT ¼ V½"H =ðV½"T þ V½"H Þ
Dividing numerator and denominator by V["T]V["H] yields wT ¼ V½"T 1 = V½"T 1 þ V½"H 1 and, consequently, wH ¼ V½"H 1 = V½"T 1 þV½"H 1 The last two formulae imply that the weights should be chosen inversely proportional to the variances of errors of both contributing estimates. The standard deviations of errors computed from the data were 1.35 and 0.97 years, respectively, and, hence, wT ¼ 1:352 = 1:352 þ0:972 ¼ 0:34 wH ¼ 0:972 = 1:352 þ0:972 ¼ 0:66 Aa ¼ 0:34 AT þ0:66 AH The relationship between ages estimated as 0.34AT + 0.66AH and chronological age of subjects whose
Combining methods of age estimation
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DOI: 10.3109/03014460.2015.1046487
Figure 3. Combined age estimates plotted against chronological age in 264 subjects whose panoramic and hand radiographs were obtained within 1 month. In regard of the diagnosis of passing the age limit of 14 years, data points in the upper left quadrant defined by the dashed lines represent false negative cases and false positives are in the lower right quadrant.
both radiographs were obtained within 1 month is illustrated in Figure 3. The standard deviation of the errors of the aggregated estimate equals 1=2 SD½"a ¼ V½"T 1 þV½"H 1 Inserting the standard deviations computed from the data yields 1=2 ¼ 0:79½years SD½"a ¼ 1:352 þ 0:972 Means and standard deviations of the errors of each method are summarised in Table 2. In order to statistically demonstrate the better performance of the combined method, we analysed in how many cases the combination improved or worsened (in terms of the absolute error) age estimation, compared to the use of only the hand or third molar method. Performance was improved more frequently than worsened (see Table 2). Case example Suppose a violent subject of unproven age claims to be 13 years 9 months old. The estimated age was 16.5 years from third molars and 15.5 years from the hand. The forensic expert should provide a conclusion about the subject’s criminal responsibility (age limit 14 years) and about the credibility of the subject’s own statement on age, taking into account the age estimates obtained from both methods. According to the preceding computations, the aggregated age estimate is 0.3416.5 + 0.6615.5 ¼ 15.84 years. This estimate is 2.33 SD (of errors) above the legally relevant age of 14 years and 2.65 SD above the claimed age of 13.75 years. Thus, the probabilities that an individual aged exactly
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14 or 13.75 years would be estimated to be 15.84 years old are 0.01 or 0.004, respectively. In other words, if an estimated age of 15.84 years or above would be accepted as a diagnosis of criminal responsibility, less than one out of 100 delinquents aged under 14 years and less than 0.4% of those under 13.75 years would be prosecuted in error. It is now the part of the court to decide whether such a rate of false positive decisions is acceptable after consideration of all the circumstances of the particular offence. Using a single age estimation method the respective probabilities would be 0.032 and 0.021 for the third molar method and 0.061 and 0.036 for the hand method. Figure 4 provides an impression of the effect of combining the two methods in the general case. It depicts the probabilities of a false positive diagnosis of an age above 14 years. The risks of errors when a single method is used are the values on the axes. The five curves in the diagram represent five levels of the error rate of the combined method. They illustrate which combinations of error rates of the hand and the third molar method will result in the respective error rate of the combined method. The case from our example is represented by the dashed lines.
Discussion Various methods of age estimation have been developed, exploring different parts of the body which undergo changes during ageing. The precision of each method in concluding the probable chronological age from developmental stages of biological characteristics is limited due to biological variability of the ages at which processes of development are initiated and due to varying rates of maturation. The only way to overcome this limitation is by combining several methods. If the acceleration or delay of the involved biological characteristics is not in perfect agreement, their combination may lead to age estimates with a higher precision than estimates from a single method in the sense that the variance of the errors will be smaller. This is particularly important in forensic age estimation in living subjects when assessing the likelihood of being on one or either side of an age threshold. Several methods of age estimation are routinely being applied in forensic age estimation; for example, in a criminal proceeding (Schmeling et al., 2008). However, there are no guidelines or recommendations so far as to how estimates from several methods should be combined into a summary statement. Here we present a strict mathematical procedure combining the age estimates from third molars and the hand and demonstrate numerically the increase in precision of the combined estimate in comparison to the estimates obtained from single methods. The key for the combination of the two estimates was the examination of the correlation between the estimation errors. When two methods have identical errors, they provide identical estimates of age and, hence, application of a second method would add no further information. The higher the correlation between two errors, the greater the similarity between the methods and the less information is gained when two methods instead of only one are applied. On the other hand, if the errors are poorly correlated then each of the methods substantially contributes its own information and
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Table 2. Distribution of errors of the methods for dental and skeletal age and their combination. Comparison of absolute errors (vs combined method) Method
Mean error [95% CI]
SD of errors
4Combined n (%)
5Combined n (%)
p Value
Third molars Hand Combined
0 (by construction) 0.37 [0.47; 0.27] 0.26 [0.32; 0.17]
1.35 0.97 0.79
275 (72%) 230 (60%) –
108 (28%) 153 (40%) –
50.001 50.001 –
The estimation error is, hence, determined by four independent components:
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" ¼ estimation error ¼ estimated agetrue age ¼ AD þ EM þ EQ þ ER
Figure 4. Illustration of the rate of false positive diagnosis of surpassing the age of 14 years when using the combined estimate, depending on the error rates of the particular methods. The dashed lines represent the scenario of the case example.
their combination will improve the overall precision of age estimation. The best situation is that errors are uncorrelated, because then each of the methods carries a maximum of information independent from the other and the combination of both is most gainful. This is the scenario that is demonstrated for Demirjian’s assessment of third molars and Thiemann’s assessment of the hand. Some notes should be made to understand this result and the requirements for its application. Several factors contribute to the age estimated from the assessment of biological characteristics: estimated age ¼ true age þ AD þ EM þ EQ þ ER Here, AD denotes the acceleration or delay of the individual relative to the population average. EM represents the error of method which may contain a bias but also other components of variance resulting from the imperfectness of the method, for example, when a formula does not use all the information that the biological characteristic under consideration could provide. EQ denotes errors due to the quality of assessment which includes poor image quality, observer bias and observer imprecision. ER reflects the random error which is not reproducible in a repeat measurements design.
The variance caused by EM can possibly be reduced by continuous re-evaluation of existing formulae. Recent considerations demonstrate that even well-established methods may have some potential for further improvement (Gelbrich et al., 2015). Training and supervision may keep the EQ component of variance as low as possible. The selection of high-quality radiographs for a study may also contribute. The variance of the component ER can be reduced by averaging over multiple assessments by either a couple of well-trained observers or by the same observer with an appropriate time gap. However, if the contribution of ER to the overall variance is small the potential gain in reducing variance of errors is small too, and must be reasonably weighted against the effort of repeat assessments. Eventually, the component of variance caused by AD cannot be reduced at all when using a single method of age estimation, since biological variability cannot be switched off. However, when combining two (or more) methods and their AD components are not in perfect agreement, then over-estimation of age by one method may counterbalance under-estimation by the other and the variance of estimation errors can be reduced beyond the biological variance that determines the limit of a single method. The present work examined the correlation of the total estimation errors of the teeth-based method, ADT + EMT + EQT + ERT, and of the hand-based method, ADH + EMH + EQH + ERH. Since there is no methodological link between both, we can assume that EMT and EMH are independent of each other. There should also be no dependency between the quality of panoramic images and hand X-rays. Furthermore, the assessments of the teeth and hand radiographs were carried out by different observers and none of them had any knowledge of classifications provided by the other. Therefore, EQT and EQH, as well as ERT and ERH, can be assumed to be independent. As a consequence, any correlation between the total estimation errors could be attributed to a correlation of ADT and ADH, i.e. a joint acceleration or delay of dental and skeletal development. Vice versa, lack of correlation of the total estimation errors implies that ADT and ADH are uncorrelated. To justify this conclusion, it is important that the assessments have sufficient quality. Otherwise EQT and EQH could contribute large uncorrelated components of variance that would drive the correlation of total estimation
DOI: 10.3109/03014460.2015.1046487
errors towards zero, although ADT and ADH were correlated. As we can suppose that only ADT and ADH contribute to the correlation of estimation errors, the correlation coefficient equals
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r ¼ CovðADT , ADH Þ=ðV½"T V½"H Þ1=2 In the analysis of repeat assessments by different observers (n ¼ 126 for third molars, n ¼ 242 for the hand) we calculated that observer effects contribute 12% and 28% to the variance of errors of the third molar and the hand method, respectively. These quantities reflect the joint contributions of intra- and inter-observer variances. When entirely eliminating the observer effects (which is practically infeasible), the variances of the errors V["T] and V["H] would decrease by factors 0.88 and 0.72, respectively, which would result in an increase of r by factor 1.25. As the upper 95% confidence limit of r was 0.08, we could believe in a true correlation coefficient of 0.10 or less. This demonstrates that the EQ components were obviously not responsible for the disappearance of a meaningful correlation. Another point to consider is the issue of independency of ADT and ADH. It is known in statistics that independent variables are uncorrelated but lack of correlation does not, in general, imply independency. However, if there was a dependency between ADT and ADH one would imagine a joint mechanism driving acceleration or delay of teeth and bones concordantly. There is no biological reasoning for another type of dependency. If this assumption is accepted, a dependency should result in a positive correlation of age estimation errors of the two methods considered. The null correlation of the estimation errors may, therefore, be interpreted as lack of dependency between both. This implies that the only joint driver of dental and skeletal age is chronological age. Individual accelerations or delays (relative to the population average) of dental and skeletal development occur independently from each other. This result has an important implication for the creation of and use of reference data-sets. According to the law in Germany and many other countries, a clinical indication is needed to obtain a radiograph from an individual. As a consequence, only such radiographs can be used to generate reference data. For the combination of two methods it would be necessary to have clinical indications for two different types of radiography, for example, a panoramic image of the teeth and a hand X-ray, nearly at the same time. This is the case in the context of an orthodontic treatment and, hence, orthodontic departments are the ideal source of data for such reference data-sets. However, it is much easier to obtain separate reference data-sets for the teeth (using data of any patients of dental practices without the restriction to orthodontic treatment) and the hand (data from emergency and surgical departments could be included here). Moreover, the recommendations for the forensic age estimation (Schmeling et al., 2008) suggest the use of reference data of the appropriate ethnicity. Availability of data may vary depending on the region, so it may sometimes be difficult to obtain reference data for separate methods and nearly impossible to get joint data of teeth and hands. Thanks to the present results, the latter is
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not necessary. Dental and skeletal age can be estimated using distinct reference data-sets and both can be combined straightforwardly. It is of paramount importance in our considerations that the assessments of teeth and hand were carried out by independent observers. The current practice in legal proceedings is committing a single forensic expert who carries out all examinations and assessments and provides the summary conclusion about age diagnostics. It is not hard to understand that the expert might prefer the particular age estimates obtained from teeth and hand to be quite consistent, so the evaluation of both may influence each other. Mathematically speaking, assessment by a single observer might create a positive correlation between EQT and EQH and the assumption of uncorrelated estimation errors would no longer be true. Therefore, to justify the application of a combination of the two methods as described in the results section, it is recommended that age estimations based on third molars and hand images are provided by independent experts and summarised in a final age diagnosis by a co-ordinating expert. Obviously, this procedure would be more costly than the involvement of a single evaluator, but it seems to be the only way to ensure the correctness of calculated probabilities. Note that the independency of estimation errors reported here cannot be generalised ad hoc to any two methods of age estimation. If acceleration or delay of skeletal development would be influenced by a systemic driving mechanism, two methods exploring bone age might have correlated estimation errors. According to current practice of forensic age estimation, third molars and the clavicular epiphysis are examined when the case is about the age limit of 18 or 21 years, so the correlation of estimation errors of these two methods would be of great interest. However, obtaining an appropriate data-set of living individuals with known ages to study this matter is difficult as participating subjects should have clinical indications for both radiographs within a short time interval. Some comment should be made with regard to the probabilities presented in the case example. We discussed the probability to estimate a subject to be aged 15.84 years or older when assuming a true age of 14 years. In terms of diagnostics, the risk of a false positive diagnosis, given the negative ‘‘true’’ state of the individual, is the lack of specificity of the diagnostic test. Instead, if one wishes to state the probability of a true age given the observation (i.e. the predictive value of the test), the Bayesian perspective should be taken. The Bayesian likelihood function L(AH, ATjage) is the density function of the probability to observe the hand bone age AH and the third molar age AT given the true age. In the context of forensic age estimation, the Bayesian prior should be a uniform distribution on an interval [14–d, 14 + d] centred at 14 years. Such a prior reflects the objectivity of the forensic expert: he or she should have no a priori preference of specific ages (uniform distribution) and should be unbiased, i.e. ages below and above the critical limit are considered to be equally likely (the interval is centred at 14 years). Restriction to an interval of 14 ± d years is justified since subjects who are apparently too young or too old would not have to undergo forensic age diagnosis
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in practice. The probability of an age below 14 years given the observation is then computed as Prob ½age 5 14 j AH , AT Z Z ¼ LðAH , AT jxÞ dx= ½14d, 14
LðAH , AT jxÞ dx ½14d, 14þd
Since acceleration or delay of hand bones and third molars are supposed to be independent, the joint likelihood can be computed as the product of the likelihood functions for each method:
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LðAH , AT jxÞ¼ LðAH jxÞ LðAT jxÞ Assuming that children younger than 10 years and adults (older than 18 years) would not at all be subject to the diagnosis of the age of 14 years since the case is obvious from their appearance, d ¼ 4 would be a reasonable choice. With our data, we computed Bayesian posterior probability that a subject with AH ¼ 15.5 and AT ¼ 16.5 years was aged 514 years to be 0.005. For the separate methods these probabilities were 0.038 (hand) and 0.036 (third molars). Four limitations of the present work should be discussed. First, the null hypothesis of no correlation between estimation errors can never be proved. The confidence interval of the correlation coefficient can only ensure that there is probably no meaningful correlation, as indicated by the confidence limits. It might be desirable to have more stringent confidence limits. The result should, therefore, be confirmed in a larger sample. The second limitation concerns the sample cohort consisting of patients of orthodontic departments which is obviously not representative for the entire population. However, in regard of the legal requirements for the indication to obtain radiographs, there is no other option than using such patient samples. This practice is widely accepted and subjects with syndromes affecting development were excluded. Orthodontic patients without syndromes can perhaps be considered to be representative for the population in regard of tooth and bone mineralisation and there is no reason to suspect the results obtained in this cohort were biased. A third limitation is that our dental age formula and the weights for combining dental and skeletal age were fitted to our sample. Therefore, the variance of errors does not incorporate the variance arising from the estimation of the coefficients of the formulae (which is part of the EM component of the errors). When our formulae were evaluated in an independent test sample, overall errors will, therefore, have larger variances than those reported herein. Furthermore, when a larger number of observers are involved who did not undergo identical training, between-observer variances may be higher than found in our observers who got the same instructions by their supervisors. Note, however, that our goal was examining the correlation between errors of two methods and demonstrating the principle of combining age estimates, not establishing a new formula. For the latter purpose, larger samples would be appropriate. A last note should be made on estimating dental age from third molars. This was motivated by the fact that legally
relevant age limits are 14 years or higher in Germany. Only third molars are then suitable for age diagnostics since the rest of the dentition is already completely mineralised in too many children before the age of 14 years and, hence, these teeth are unspecific for the detection of an age above 14 years. In countries where the age of criminal responsibility is 10 years, dental age would not be estimated from third molars but from the other teeth, since they can discriminate between ages below and above 10 years and deliver much more precise information. Our results should not ad hoc be generalised to this setting.
Conclusion Strict algorithms for the combination of age estimates obtained from several methods exploring different body systems are needed. The present work shows that age estimation methods using the developing third molars and the hand have independent estimation errors. This allows straightforward computation of an aggregated age estimate. Application of the formula that combines the two methods requires independency of the errors of the particular age estimates. It is, therefore, recommended that the age estimates from teeth and hand are provided by independent experts. The results also imply that separate reference samples for the third molar and hand can be established and the methods can, nonetheless, be combined; no joint reference data of teeth and hand are necessary. The combination of other methods of age estimation, for example methods using third molars and the clavicular epiphysis, still need to be investigated.
Acknowledgements This paper is dedicated to the memory of Dr Klaus Ro¨tzscher (Speyer, Germany) who introduced the first and the last author into the community of forensic scientists and invited them to present their interdisciplinary contributions on the forensic meetings. Unfortunately, he died on 21 October 2014 and, therefore, could not see the newest results of the work he had so warmly encouraged. The authors also wish to express their gratitude to the anonymous reviewers whose fruitful comments and suggestions significantly contributed to improving this paper.
Declaration of interest All authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.
References Demirjian A, Goldstein H, Tanner JM. 1973. A new system of dental age assessment. Hum Biol 45:211–227. Gelbrich B, Schwerdt S, Hirsch A, Dannhauer KH, Tausche E, Gelbrich G. 2015. Various faces of age estimation: methodological considerations from the perspective of individual disciplines. Rechtsmedizin 25:7–11. Nolla C. 1960. The development of permanent teeth. J Dent Child 27: 254–266. Sachs L. 1997. Applied statistics. 8th ed. Berlin: Springer. [German]. Schmeling A, Grundmann C, Fuhrmann A, Kaatsch HJ, Knell B, Ramsthaler F, Reisinger W, et al. 2008. Updated recommendations of the Study Group on Forensic Age Diagnostics for age estimation in living individuals in criminal proceedings. Rechtsmedizin 18: 451–453. [German]. Thiemann HH, Nitz I, Schmeling A. 2006. Radiographic atlas of the normal hand in childhood. 3rd ed. Stuttgart: Thieme. [German].
RESEARCH PAPER
Combining wrist age and third molars in forensic age estimation: how to calculate the joint age estimate and its error rate in age diagnostics* Bianca Gelbrich1, Carolin Frerking1, Sandra Weiß2, Sebastian Schwerdt1, Angelika Stellzig-Eisenhauer2, Eve Tausche3, and Go¨tz Gelbrich4,5 1
Department of Orthodontics, University Hospital Leipzig, Germany, 2Department of Orthodontics, University Hospital Wu¨rzburg, Germany, Department of Orthodontics, University Hospital Dresden, Germany, 4Institute of Clinical Epidemiology and Biometry, University of Wu¨rzburg, Germany, and 5Clinical Trial Centre, University Hospital Wu¨rzburg, Germany
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Abstract
Keywords
Background: Forensic age estimation in living adolescents is based on several methods, e.g. the assessment of skeletal and dental maturation. Combination of several methods is mandatory, since age estimates from a single method are too imprecise due to biological variability. The correlation of the errors of the methods being combined must be known to calculate the precision of combined age estimates. Aim: To examine the correlation of the errors of the hand and the third molar method and to demonstrate how to calculate the combined age estimate. Subjects and methods: Clinical routine radiographs of the hand and dental panoramic images of 383 patients (aged 7.8–19.1 years, 56% female) were assessed. Results: Lack of correlation (r ¼ –0.024, 95% CI ¼ –0.124 to + 0.076, p ¼ 0.64) allows calculating the combined age estimate as the weighted average of the estimates from hand bones and third molars. Combination improved the standard deviations of errors (hand ¼ 0.97, teeth ¼ 1.35 years) to 0.79 years. Conclusion: Uncorrelated errors of the age estimates obtained from both methods allow straightforward determination of the common estimate and its variance. This is also possible when reference data for the hand and the third molar method are established independently from each other, using different samples.
Age estimation error, combined methods, dental panoramic radiograph, forensic age estimation, hand radiograph
Introduction The application of a law may depend on the age of the subject concerned. In Germany, for example, the age of criminal responsibility is 14 years. Criminal law relating to young offenders must be applied if the delinquent has not yet reached the age of 18 years and can be applied if the delinquent is still under the age of 21. Beyond criminal proceedings, young immigrants under 18 years of age deserve particular protection by the German law. The same or similar rules are applied in many other countries. In cases where age-dependent legal action is due and the age of the subject under consideration cannot be verified (for example if a young immigrant does not possess identifying documents), the authority needs to assume an age. The assumption is made on the basis of forensic expertise. Forensic age estimation explores the maturation stage of several biological structures. According to current
This paper is dedicated to the memory of Klaus Ro¨tzscher. Correspondence: Dr. Bianca Gelbrich, Department of Orthodontics, University Hospital Leipzig, Liebigstraße 10-14, Building 1, D-04103 Leipzig, Germany. Tel: +49-341-97 21 050. Fax: +49-341-97 21 059. E-mail: [email protected]
History Received 23 April 2015 Accepted 27 April 2015 Published online 16 June 2015
recommendations (Schmeling et al., 2008), the development of the hand assessed from a radiograph, mineralisation of third molars assessed from a panoramic image and the closure of the clavicular epiphysis assessed from computed tomography should be incorporated when the age of living individuals is estimated in the context of criminal proceedings. Apart from the latter, German law allows X-ray imaging only in the presence of a clinical indication. Depending on the country and national law, the forensic methods may then be limited to assessing the eruption of the third molars or, in rare cases, exploring a single radiograph when available from a medical treatment. The timing of bone calcification and mineralisation of teeth shows large biological variability. Developmental stages of subjects of the same age may be very different and, vice versa, individuals with identical mineralisation of a certain bone or tooth may differ in age by 4 years or more. As a consequence, the estimation errors (differences between estimated and true chronological ages) have a substantial variance when age would be estimated from a single biological characteristic and age diagnostics (i.e. the decision whether a certain age limit has been passed) would suffer from unacceptable error rates. Therefore, several biological structures need to be assessed and the information from
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different estimation methods should be weighed against each other and summarised in the forensic expertise. In common practice separate age estimates and measures of uncertainty (e.g. standard deviations) are quoted for each method and the overall estimate of the most probable age and statements with regard to the age limits relevant in the respective legal context are derived by discussion rather than computation. There is no established strict procedure for the aggregation of the particular estimates and the determination of the variance of the pooled age estimate. The latter would be of particular interest as it determines the precision of forensic age diagnostics. Consider two methods of age estimation, suppose their errors have equal variances, denoted by V, and let r be the correlation coefficient of the errors of both methods. Then the errors of the average of the two age estimates has the variance V (1 + r)/2. If, for example, the left and the right hand would be examined, one can assume that the skeletal age is accelerated or delayed for both hands equally or almost equally, so r would be close to 1, and the variance of the averaged age estimates would be almost as large as V. This means that, given the information from one hand, the examination of the other does not add much and does not materially contribute in improving the precision of the estimated age. If, in another scenario, acceleration or delay of the development of two biological characteristics were independent from each other, the resulting two estimates of age would have uncorrelated errors. This would be the ideal case since each method would add independent information to the other and the variance of the averaged age estimates would be equal to V/2. For the common interpretation of age estimates from different methods it is, therefore, crucial to have information about the correlation between the estimation errors. The aim of this work is to study the correlation between the errors of the age estimates obtained from the hand bones and from the mineralisation of third molars.
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Methods
more precise information on age in younger children, mineralisation of this part of the permanent dentition is finished at the age of 14 years in about half of the population. Therefore, in the setting of forensic age estimation in living subjects within legal proceedings, incomplete mineralisation of these teeth would imply sufficient doubt that the subject under assessment was older than 14 years. On the other hand, complete mineralisation would not be used to derive any statement regarding the question of whether the subject has or has not passed the threshold of 14 years or another legally relevant age limit. Hence, only third molars are helpful in the diagnosis of, for example, criminal responsibility. The mineralisation stages of third molars were assessed according to the scheme of Demirjian et al. (1973). Eight stages are defined in this classification: calcified cusp tips (A), connected occlusal surface (B), complete formation of enamel in the occlusal surface (C), completion of crown formation (D), pulp horns and pulp chamber differentiated, radicular bifurcation (E), root length starts exceeding crown height, funnel shaped apices (F), root almost complete, apices still open (G) and apices closed (H). For computations, these stages were enumerated from 1–8. Following an idea from Nolla’s (1960) classification, we added the stage of a crypt without visible mineralisation, enumerated as 0. This enables better distinguishing between teeth which are present and will start mineralising soon and teeth that are not visible at all and might develop later or even never due to aplasia. The classification was carried out by the second author after training by the first. For quality assurance, the first author re-evaluated independently a randomly chosen part of the panoramic images. In addition, statistical monitoring was performed by the last author. For conversion of Demirjian stages into estimated ages, a regression model was applied (see statistics below). We did not use established formulae as we extended Demirjian’s scheme by the 0 stage. This stage was not included in the published assessment schemes.
Subjects and materials
Estimation of skeletal age
We retrospectively evaluated routine X-ray images of patients of the orthodontic departments of the University Hospitals of Leipzig, Dresden and Wu¨rzburg, Germany. Subjects were eligible if they had a radiograph of their hand and a panoramic dental radiograph in their files, the time difference between the dates of both images was not greater than 6 months, the development of at least one third molar was visible on the panoramic image and the subject had no syndrome known to be associated with a delay of the general, skeletal or dental maturation. Cases of both hypodontia, supernumerary teeth and preceding tooth extractions were also excluded. Radiographs were eligible if they had sufficient quality to perform the assessments described below. If more than one pair of radiographs was available from a subject, the one with the smaller difference of dates was selected. In cases of equality, the radiographs of better quality were preferred.
The age estimates from the hand X-rays were obtained by the atlas method of Thiemann et al. (2006). The atlas handbook contains a series of reference radiographs of the left hand, one for each age group and sex, accompanied by a description of the developmental features of the hand bones which are characteristic for the respective age. The observer should iteratively find the reference chart that best fits the image under assessment and this determines the estimated age. The assessment was carried out by the third and the fourth author (for the images from Leipzig/Dresden and Wu¨rzburg, respectively). Again, the first and last author provided training and supervision.
Estimation of dental age Dental age was estimated from the development of third molars. Although incisors through second molars provide
Statistics Estimated dental age was calculated by a multiple linear regression model, adjusting for sex and the structure of missing third molars. The estimation errors were computed as the differences between estimated ages and chronological age. The relationship between both estimation errors was assessed by Pearson’s correlation coefficient. Its 95% confidence
Combining methods of age estimation
DOI: 10.3109/03014460.2015.1046487
interval (CI) was computed using Fisher’s transform (Sachs, 1997, pp. 541–542). SPSS 22 (IBM Corp., Armonk, NY) was used as statistical software. Ethics Since there were no experimental issues, data were obtained retrospectively from clinical routine images and data transferred for analysis were anonymous, neither ethical approval nor informed consent of the subjects was necessary according to legal requirements in Germany. Nonetheless, the study protocol defining the research project, which the present work is part of, has been approved by the Ethics Committee of the Medical Faculty of Leipzig University, Germany.
Results Ann Hum Biol Downloaded from informahealthcare.com by University of Otago on 07/15/15 For personal use only.
Sample characteristics The sample included 383 subjects, 213 girls (56%) and 170 boys. Ages ranged from 7.8–19.1 years. Frequencies of age groups and sexes are displayed in Table 1. The radiographs of 264 subjects (69%) were obtained within 1 month and in a further 80 subjects (21%) the dates of the radiographs were more than 1 month but no more than 3 months apart from each other. Estimated dental age The regression formula for age estimation from third molars, taking into account the configuration of present third molars and the sex of the subject, yielded: Estimated age ðyearsÞ ¼ 7:79 þ 1:26 average Demirjianstage of third molars þ 0:54 ½third molars incomplete AND ðN Max 4 N Mand Þ þ1:27 ½third molars incomplete AND ðN Max N Mand Þ 0:16 ½female sex where NMax and NMand denote the numbers of present maxillary and mandibular third molars, respectively. Each of the three logical expressions in brackets takes the numerical value 1 if true and 0 if false. The coefficients and their 95% CI were: intercept 7.79 (7.33–8.24), slope 1.26 (1.16–1.36), shifts for at least one missing third molar 0.54 (0.09–0.98) if NMax4NMand and 1.27 (0.87–1.66) if NMax NMand. The shift for female sex (0.16, 0.44 to +0.11) was not formally significant (p ¼ 0.25), but was left in the model since a slightly earlier development of third molars in girls (our estimate yields 2 months) is plausible. Table 1. Distribution of age and sex in the sample. Age group (years) 7.8–59 9–510 10–511 11–512 12–513 13–514 14–515 15–516 16–19.1
n (boys)
n (girls)
3 18 29 23 32 28 16 13 8
15 22 38 42 34 23 14 14 11
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Since this regression model was fitted to the data, there is no bias by construction. The standard deviation of the residuals, i.e. the estimation errors, was 1.35 years. The relationship between age estimates obtained from third molars and chronological age is illustrated in Figure 1(A). Estimated skeletal age The estimates of age from hand radiographs according to the Thiemann-Nitz method were slightly biased, the mean error was 0.37 years (95% CI ¼ –0.47 to 0.27). This corresponds to an under-estimation of age by 4 months, which is in favour of the subjects under forensic assessment. The standard deviation of the estimation errors was 0.97 years. The relationship between age estimates obtained from the hand and chronological age is illustrated in Figure 1(B). Correlation of estimation errors The correlation coefficient of the estimation errors of both methods was 0.024 (95% CI ¼ –0.124 to +0.076, p ¼ 0.64). Figure 2(A) illustrates the lack of correlation. Selecting only the 264 subjects whose radiographs were obtained within 1 month yielded a correlation coefficient of 0.007 (95% CI ¼ –0.128 to + 0.114, p ¼ 0.91). Looking at Figure 2, we can divide the errors of both methods into tertile groups. Let T1, T2, T3 and H1, H2, H3 denote the lowest, middle and highest tertile groups of the estimation errors of the third molar and hand method, respectively (Figure 2B). One might expect a positive association in the sense that, for example, when dental age is accelerated it would be more likely that the skeletal age is also accelerated rather than delayed. In other words, within the group T3 more subjects belonging to H3 than to H1 would have been expected. Vice versa, within T1 one would have expected an over-representation of H1 and an under-representation of H3. This was not the case. The lack of correlation is reflected by the nearly equal numbers of subjects in all nine combinations of tertiles. Aggregated estimated age The aggregated estimate of age Aa is computed as the weighted average of ages estimated from the third molars (AT) and the hand (AH): Aa ¼ wT AT þwH AH The respective weights wT and wH are non-negative numbers fulfilling wT + wH ¼ 1. We now derive how wT and wH should be chosen to minimise the variance of errors "a of the aggregated estimate. The errors are equal to "a ¼ wT AT þ wH AH Ac ¼ wT ðAc þ"T Þ þ wH :ðAc þ"H Þ Ac ¼ wT :"T þ wH :"H where Ac is the chronological age and "T, "H are the errors of the age estimates obtained from teeth and
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Figure 2. (A) Scatter plot of the errors of ages estimated from hand (horizontal axis) and teeth (vertical axis). The regression line illustrates the lack of a meaningful correlation. (B) Frequencies of subjects in the tertile groups of the estimation errors. The combinations of tertiles of both methods occur nearly equally often.
Figure 1. Ages estimated from third molars (A) and the hand (B) plotted against chronological age. In regard of the diagnosis of passing the age limit of 14 years, data points in the upper left quadrant defined by the dashed lines represent false negative cases and false positives are in the lower right quadrant.
hand, respectively. Thanks to the null correlation of the errors "T and "H and applying wH ¼ 1 – wT, the variance of "a computes as V½"a ¼ w2T V½"T þð1wT Þ2 V½"H ¼ ðV½"T þV½"H Þ w2T 2 V½"H wT þV½"H This quadratic function in wT attains its minimum at the zero of its derivative d=dwT V½"a ¼ 2 ðV½"T þ V½"H ÞwT 2 V½"H ¼ 0 which implies wT ¼ V½"H =ðV½"T þ V½"H Þ
Dividing numerator and denominator by V["T]V["H] yields wT ¼ V½"T 1 = V½"T 1 þ V½"H 1 and, consequently, wH ¼ V½"H 1 = V½"T 1 þV½"H 1 The last two formulae imply that the weights should be chosen inversely proportional to the variances of errors of both contributing estimates. The standard deviations of errors computed from the data were 1.35 and 0.97 years, respectively, and, hence, wT ¼ 1:352 = 1:352 þ0:972 ¼ 0:34 wH ¼ 0:972 = 1:352 þ0:972 ¼ 0:66 Aa ¼ 0:34 AT þ0:66 AH The relationship between ages estimated as 0.34AT + 0.66AH and chronological age of subjects whose
Combining methods of age estimation
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DOI: 10.3109/03014460.2015.1046487
Figure 3. Combined age estimates plotted against chronological age in 264 subjects whose panoramic and hand radiographs were obtained within 1 month. In regard of the diagnosis of passing the age limit of 14 years, data points in the upper left quadrant defined by the dashed lines represent false negative cases and false positives are in the lower right quadrant.
both radiographs were obtained within 1 month is illustrated in Figure 3. The standard deviation of the errors of the aggregated estimate equals 1=2 SD½"a ¼ V½"T 1 þV½"H 1 Inserting the standard deviations computed from the data yields 1=2 ¼ 0:79½years SD½"a ¼ 1:352 þ 0:972 Means and standard deviations of the errors of each method are summarised in Table 2. In order to statistically demonstrate the better performance of the combined method, we analysed in how many cases the combination improved or worsened (in terms of the absolute error) age estimation, compared to the use of only the hand or third molar method. Performance was improved more frequently than worsened (see Table 2). Case example Suppose a violent subject of unproven age claims to be 13 years 9 months old. The estimated age was 16.5 years from third molars and 15.5 years from the hand. The forensic expert should provide a conclusion about the subject’s criminal responsibility (age limit 14 years) and about the credibility of the subject’s own statement on age, taking into account the age estimates obtained from both methods. According to the preceding computations, the aggregated age estimate is 0.3416.5 + 0.6615.5 ¼ 15.84 years. This estimate is 2.33 SD (of errors) above the legally relevant age of 14 years and 2.65 SD above the claimed age of 13.75 years. Thus, the probabilities that an individual aged exactly
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14 or 13.75 years would be estimated to be 15.84 years old are 0.01 or 0.004, respectively. In other words, if an estimated age of 15.84 years or above would be accepted as a diagnosis of criminal responsibility, less than one out of 100 delinquents aged under 14 years and less than 0.4% of those under 13.75 years would be prosecuted in error. It is now the part of the court to decide whether such a rate of false positive decisions is acceptable after consideration of all the circumstances of the particular offence. Using a single age estimation method the respective probabilities would be 0.032 and 0.021 for the third molar method and 0.061 and 0.036 for the hand method. Figure 4 provides an impression of the effect of combining the two methods in the general case. It depicts the probabilities of a false positive diagnosis of an age above 14 years. The risks of errors when a single method is used are the values on the axes. The five curves in the diagram represent five levels of the error rate of the combined method. They illustrate which combinations of error rates of the hand and the third molar method will result in the respective error rate of the combined method. The case from our example is represented by the dashed lines.
Discussion Various methods of age estimation have been developed, exploring different parts of the body which undergo changes during ageing. The precision of each method in concluding the probable chronological age from developmental stages of biological characteristics is limited due to biological variability of the ages at which processes of development are initiated and due to varying rates of maturation. The only way to overcome this limitation is by combining several methods. If the acceleration or delay of the involved biological characteristics is not in perfect agreement, their combination may lead to age estimates with a higher precision than estimates from a single method in the sense that the variance of the errors will be smaller. This is particularly important in forensic age estimation in living subjects when assessing the likelihood of being on one or either side of an age threshold. Several methods of age estimation are routinely being applied in forensic age estimation; for example, in a criminal proceeding (Schmeling et al., 2008). However, there are no guidelines or recommendations so far as to how estimates from several methods should be combined into a summary statement. Here we present a strict mathematical procedure combining the age estimates from third molars and the hand and demonstrate numerically the increase in precision of the combined estimate in comparison to the estimates obtained from single methods. The key for the combination of the two estimates was the examination of the correlation between the estimation errors. When two methods have identical errors, they provide identical estimates of age and, hence, application of a second method would add no further information. The higher the correlation between two errors, the greater the similarity between the methods and the less information is gained when two methods instead of only one are applied. On the other hand, if the errors are poorly correlated then each of the methods substantially contributes its own information and
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Table 2. Distribution of errors of the methods for dental and skeletal age and their combination. Comparison of absolute errors (vs combined method) Method
Mean error [95% CI]
SD of errors
4Combined n (%)
5Combined n (%)
p Value
Third molars Hand Combined
0 (by construction) 0.37 [0.47; 0.27] 0.26 [0.32; 0.17]
1.35 0.97 0.79
275 (72%) 230 (60%) –
108 (28%) 153 (40%) –
50.001 50.001 –
The estimation error is, hence, determined by four independent components:
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" ¼ estimation error ¼ estimated agetrue age ¼ AD þ EM þ EQ þ ER
Figure 4. Illustration of the rate of false positive diagnosis of surpassing the age of 14 years when using the combined estimate, depending on the error rates of the particular methods. The dashed lines represent the scenario of the case example.
their combination will improve the overall precision of age estimation. The best situation is that errors are uncorrelated, because then each of the methods carries a maximum of information independent from the other and the combination of both is most gainful. This is the scenario that is demonstrated for Demirjian’s assessment of third molars and Thiemann’s assessment of the hand. Some notes should be made to understand this result and the requirements for its application. Several factors contribute to the age estimated from the assessment of biological characteristics: estimated age ¼ true age þ AD þ EM þ EQ þ ER Here, AD denotes the acceleration or delay of the individual relative to the population average. EM represents the error of method which may contain a bias but also other components of variance resulting from the imperfectness of the method, for example, when a formula does not use all the information that the biological characteristic under consideration could provide. EQ denotes errors due to the quality of assessment which includes poor image quality, observer bias and observer imprecision. ER reflects the random error which is not reproducible in a repeat measurements design.
The variance caused by EM can possibly be reduced by continuous re-evaluation of existing formulae. Recent considerations demonstrate that even well-established methods may have some potential for further improvement (Gelbrich et al., 2015). Training and supervision may keep the EQ component of variance as low as possible. The selection of high-quality radiographs for a study may also contribute. The variance of the component ER can be reduced by averaging over multiple assessments by either a couple of well-trained observers or by the same observer with an appropriate time gap. However, if the contribution of ER to the overall variance is small the potential gain in reducing variance of errors is small too, and must be reasonably weighted against the effort of repeat assessments. Eventually, the component of variance caused by AD cannot be reduced at all when using a single method of age estimation, since biological variability cannot be switched off. However, when combining two (or more) methods and their AD components are not in perfect agreement, then over-estimation of age by one method may counterbalance under-estimation by the other and the variance of estimation errors can be reduced beyond the biological variance that determines the limit of a single method. The present work examined the correlation of the total estimation errors of the teeth-based method, ADT + EMT + EQT + ERT, and of the hand-based method, ADH + EMH + EQH + ERH. Since there is no methodological link between both, we can assume that EMT and EMH are independent of each other. There should also be no dependency between the quality of panoramic images and hand X-rays. Furthermore, the assessments of the teeth and hand radiographs were carried out by different observers and none of them had any knowledge of classifications provided by the other. Therefore, EQT and EQH, as well as ERT and ERH, can be assumed to be independent. As a consequence, any correlation between the total estimation errors could be attributed to a correlation of ADT and ADH, i.e. a joint acceleration or delay of dental and skeletal development. Vice versa, lack of correlation of the total estimation errors implies that ADT and ADH are uncorrelated. To justify this conclusion, it is important that the assessments have sufficient quality. Otherwise EQT and EQH could contribute large uncorrelated components of variance that would drive the correlation of total estimation
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errors towards zero, although ADT and ADH were correlated. As we can suppose that only ADT and ADH contribute to the correlation of estimation errors, the correlation coefficient equals
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r ¼ CovðADT , ADH Þ=ðV½"T V½"H Þ1=2 In the analysis of repeat assessments by different observers (n ¼ 126 for third molars, n ¼ 242 for the hand) we calculated that observer effects contribute 12% and 28% to the variance of errors of the third molar and the hand method, respectively. These quantities reflect the joint contributions of intra- and inter-observer variances. When entirely eliminating the observer effects (which is practically infeasible), the variances of the errors V["T] and V["H] would decrease by factors 0.88 and 0.72, respectively, which would result in an increase of r by factor 1.25. As the upper 95% confidence limit of r was 0.08, we could believe in a true correlation coefficient of 0.10 or less. This demonstrates that the EQ components were obviously not responsible for the disappearance of a meaningful correlation. Another point to consider is the issue of independency of ADT and ADH. It is known in statistics that independent variables are uncorrelated but lack of correlation does not, in general, imply independency. However, if there was a dependency between ADT and ADH one would imagine a joint mechanism driving acceleration or delay of teeth and bones concordantly. There is no biological reasoning for another type of dependency. If this assumption is accepted, a dependency should result in a positive correlation of age estimation errors of the two methods considered. The null correlation of the estimation errors may, therefore, be interpreted as lack of dependency between both. This implies that the only joint driver of dental and skeletal age is chronological age. Individual accelerations or delays (relative to the population average) of dental and skeletal development occur independently from each other. This result has an important implication for the creation of and use of reference data-sets. According to the law in Germany and many other countries, a clinical indication is needed to obtain a radiograph from an individual. As a consequence, only such radiographs can be used to generate reference data. For the combination of two methods it would be necessary to have clinical indications for two different types of radiography, for example, a panoramic image of the teeth and a hand X-ray, nearly at the same time. This is the case in the context of an orthodontic treatment and, hence, orthodontic departments are the ideal source of data for such reference data-sets. However, it is much easier to obtain separate reference data-sets for the teeth (using data of any patients of dental practices without the restriction to orthodontic treatment) and the hand (data from emergency and surgical departments could be included here). Moreover, the recommendations for the forensic age estimation (Schmeling et al., 2008) suggest the use of reference data of the appropriate ethnicity. Availability of data may vary depending on the region, so it may sometimes be difficult to obtain reference data for separate methods and nearly impossible to get joint data of teeth and hands. Thanks to the present results, the latter is
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not necessary. Dental and skeletal age can be estimated using distinct reference data-sets and both can be combined straightforwardly. It is of paramount importance in our considerations that the assessments of teeth and hand were carried out by independent observers. The current practice in legal proceedings is committing a single forensic expert who carries out all examinations and assessments and provides the summary conclusion about age diagnostics. It is not hard to understand that the expert might prefer the particular age estimates obtained from teeth and hand to be quite consistent, so the evaluation of both may influence each other. Mathematically speaking, assessment by a single observer might create a positive correlation between EQT and EQH and the assumption of uncorrelated estimation errors would no longer be true. Therefore, to justify the application of a combination of the two methods as described in the results section, it is recommended that age estimations based on third molars and hand images are provided by independent experts and summarised in a final age diagnosis by a co-ordinating expert. Obviously, this procedure would be more costly than the involvement of a single evaluator, but it seems to be the only way to ensure the correctness of calculated probabilities. Note that the independency of estimation errors reported here cannot be generalised ad hoc to any two methods of age estimation. If acceleration or delay of skeletal development would be influenced by a systemic driving mechanism, two methods exploring bone age might have correlated estimation errors. According to current practice of forensic age estimation, third molars and the clavicular epiphysis are examined when the case is about the age limit of 18 or 21 years, so the correlation of estimation errors of these two methods would be of great interest. However, obtaining an appropriate data-set of living individuals with known ages to study this matter is difficult as participating subjects should have clinical indications for both radiographs within a short time interval. Some comment should be made with regard to the probabilities presented in the case example. We discussed the probability to estimate a subject to be aged 15.84 years or older when assuming a true age of 14 years. In terms of diagnostics, the risk of a false positive diagnosis, given the negative ‘‘true’’ state of the individual, is the lack of specificity of the diagnostic test. Instead, if one wishes to state the probability of a true age given the observation (i.e. the predictive value of the test), the Bayesian perspective should be taken. The Bayesian likelihood function L(AH, ATjage) is the density function of the probability to observe the hand bone age AH and the third molar age AT given the true age. In the context of forensic age estimation, the Bayesian prior should be a uniform distribution on an interval [14–d, 14 + d] centred at 14 years. Such a prior reflects the objectivity of the forensic expert: he or she should have no a priori preference of specific ages (uniform distribution) and should be unbiased, i.e. ages below and above the critical limit are considered to be equally likely (the interval is centred at 14 years). Restriction to an interval of 14 ± d years is justified since subjects who are apparently too young or too old would not have to undergo forensic age diagnosis
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in practice. The probability of an age below 14 years given the observation is then computed as Prob ½age 5 14 j AH , AT Z Z ¼ LðAH , AT jxÞ dx= ½14d, 14
LðAH , AT jxÞ dx ½14d, 14þd
Since acceleration or delay of hand bones and third molars are supposed to be independent, the joint likelihood can be computed as the product of the likelihood functions for each method:
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LðAH , AT jxÞ¼ LðAH jxÞ LðAT jxÞ Assuming that children younger than 10 years and adults (older than 18 years) would not at all be subject to the diagnosis of the age of 14 years since the case is obvious from their appearance, d ¼ 4 would be a reasonable choice. With our data, we computed Bayesian posterior probability that a subject with AH ¼ 15.5 and AT ¼ 16.5 years was aged 514 years to be 0.005. For the separate methods these probabilities were 0.038 (hand) and 0.036 (third molars). Four limitations of the present work should be discussed. First, the null hypothesis of no correlation between estimation errors can never be proved. The confidence interval of the correlation coefficient can only ensure that there is probably no meaningful correlation, as indicated by the confidence limits. It might be desirable to have more stringent confidence limits. The result should, therefore, be confirmed in a larger sample. The second limitation concerns the sample cohort consisting of patients of orthodontic departments which is obviously not representative for the entire population. However, in regard of the legal requirements for the indication to obtain radiographs, there is no other option than using such patient samples. This practice is widely accepted and subjects with syndromes affecting development were excluded. Orthodontic patients without syndromes can perhaps be considered to be representative for the population in regard of tooth and bone mineralisation and there is no reason to suspect the results obtained in this cohort were biased. A third limitation is that our dental age formula and the weights for combining dental and skeletal age were fitted to our sample. Therefore, the variance of errors does not incorporate the variance arising from the estimation of the coefficients of the formulae (which is part of the EM component of the errors). When our formulae were evaluated in an independent test sample, overall errors will, therefore, have larger variances than those reported herein. Furthermore, when a larger number of observers are involved who did not undergo identical training, between-observer variances may be higher than found in our observers who got the same instructions by their supervisors. Note, however, that our goal was examining the correlation between errors of two methods and demonstrating the principle of combining age estimates, not establishing a new formula. For the latter purpose, larger samples would be appropriate. A last note should be made on estimating dental age from third molars. This was motivated by the fact that legally
relevant age limits are 14 years or higher in Germany. Only third molars are then suitable for age diagnostics since the rest of the dentition is already completely mineralised in too many children before the age of 14 years and, hence, these teeth are unspecific for the detection of an age above 14 years. In countries where the age of criminal responsibility is 10 years, dental age would not be estimated from third molars but from the other teeth, since they can discriminate between ages below and above 10 years and deliver much more precise information. Our results should not ad hoc be generalised to this setting.
Conclusion Strict algorithms for the combination of age estimates obtained from several methods exploring different body systems are needed. The present work shows that age estimation methods using the developing third molars and the hand have independent estimation errors. This allows straightforward computation of an aggregated age estimate. Application of the formula that combines the two methods requires independency of the errors of the particular age estimates. It is, therefore, recommended that the age estimates from teeth and hand are provided by independent experts. The results also imply that separate reference samples for the third molar and hand can be established and the methods can, nonetheless, be combined; no joint reference data of teeth and hand are necessary. The combination of other methods of age estimation, for example methods using third molars and the clavicular epiphysis, still need to be investigated.
Acknowledgements This paper is dedicated to the memory of Dr Klaus Ro¨tzscher (Speyer, Germany) who introduced the first and the last author into the community of forensic scientists and invited them to present their interdisciplinary contributions on the forensic meetings. Unfortunately, he died on 21 October 2014 and, therefore, could not see the newest results of the work he had so warmly encouraged. The authors also wish to express their gratitude to the anonymous reviewers whose fruitful comments and suggestions significantly contributed to improving this paper.
Declaration of interest All authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.
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